Finite nilpotent groups are direct products of p-groups; ie groups of p-power order so the question can easily be reduced to 'Is there a classification of (finite) p-groups?'

As I understand it the best known answer is via the 'co-class theoerems'; see http://en.wikipedia.org/wiki/P-group for more information and references.

http://en.wikipedia.org/wiki/P-group

Even the number of non-isomorphic groups of order p^n is unknown in general.

The classification of the finite p-groups up to isomorphism is almost ``the last domain'' towards solving the classification of finite groups.  Effectively, it is a vast project that has on one hand a finished character (given any p-group, one can come to understand it fairly quickly) and on the other hand cannot even begin to be approached (say, a nice dictionary of all iso types of all p-groups of higher orders is no-where near to being completed, and should probably be considered not even started).

There is no classification of finite nilpotent groups or equivalently of  finite p-groups (even, if I'm not mistaken, for those of class 2), moreover it is believed that there is no hope to classify them up to isomorphism.   One of the reasons is that they number is very large, for instance the number of groups of order 1024 is 49487365422.  A more reasonable project is to try  to divide them into families, in such a way that the behaviour of p-groups in the same family is well understood and can be studied in a unified framework.  We have for instance the coclass project, which takes the coclass of a p-group as the primary invariant (the coclass of a group of order p^n and class c is equal to n-c).  So the p-groups are divided into families according to their coclasses.   One of the  deep results in this approach is that the p-groups (p fixed) in the same coclass family can be seen as a graph which contains only finitely many infinite trees (one can find more details in the book "The structure of groups of prime power order" by C. Leedham-Green and S. Mckey).    The groups of large coclass may produce some difficulties to this project, and moreover the simplest within this framework are not classified (I mean the p-groups of coclass 1, except for p=2 or 3).  

I went through the above mentioned links in wikipedia regarding classification of p- groups , but can anyone tell me that has there been any concrete work done on classifying these groups because the techniques that have been mentioned above (eg co class ) and others don't seem to be much rigorous .

Dear Ayush,

  Yassine Guerboussa's answer is fairly detailed and correct to the best of my knowledge.  If you want more details, you should certainly read the book by Leedham-Green and McKey.  As I understand it, the book IS rigorous given the literature it leans on (and which is detailed in it's bibliography).

  Really, the issue here is simply the vast number of iso types available; at this juncture, further theorems start to take on a feeling of simply catalogue listings of the results. So, excepting some very broad categories (e.g., co-class), it is not completely clear what further specific contributions to the literature can offer that would go beyond the specific capabilities of programs like GAP or Magma to explore specific groups.

  That said, I am sure specific researchers can make more detailed arguments pointing to ``needed theory'' that is broader than what I mention above, so perhaps my sense that this corner is in some sense already reasonably explored is incorrect.

Thank you Collin Sir for the helpful insight .

My pleasure,  and best wishes for your future research!